Critically discuss the suitability of using the Black & Scholes model for pricing derivatives

Institution Title Name Date The Black Scholes model Introduction The Black scholes model is a mathematical model developed to describe the financial market and derivative investments; it shows the price variations over time of the financial instruments such as stock that can be used to determine the price of European call option. Option are financial instrument giving the holder the right to buy or sell an underlying stock or commodity at a future point in time at an agreed upon price .The model is a tool of pricing the equity option thus; it is an approach of calculating the value of stock option be it call option or put up option. The main idea that lies behind this model is that the price of an option is determined implicitly by the price of underlying stock. This model is one of the most important concepts of financial theory; it was developed by Fisher Black and Myron Scholes in 1973 and it is used widely today since it is known to be the best in determining the fair prices of option as we are going to see. (/). The Black Scholes model was not developed overnight, but it took a considerable length of time for the two professors to come up with it. Fisher Black started working on it when calculating the price of the stock warrants and it involved computing mathematical derivatives to ascertain how the rate of the discount of a warrant varies with stock price with time. The results of his work held an alarming resemblance with a known heat transfer equation, later Myron Scholes joined him and they came up with this accurate option pricing model (/b-s_model.htm). The model is divided into two sections, the first part shows the expected benefits one gets when he acquires a stock outright, this is fond by multiplying the stock price by the change in premium call with respect to the underlying stock price. The second part shows the present value of paying the exercise price on the expiration date The good thing with Back Scholes model is that it fits better to the dynamics of historic prices and pricing formulas for European options and it performs better than the earlier models, more over, until recently, there are no analytical formulas presented to tackle the European pricing option which can out do this one of Black and scholes, thus it has become the standard measure of pricing option. (). For some one to value an investment he has to evaluate its probability distribution, and to evaluate probability distribution, hedge funds and other market participants use geometric Brownian motion models, binomial models and stochastic calculus. Black scholes pricing theory has provided a new way of valuing stock option and more importantly it started a revolution in how hedge funds and other market participants think about and value financial assets. In application of Black Scholes model, there are several assumptions that should be observed for one to achieve accurate results. These assumptions are; the stock pays no dividends during the option’s life, this is an alarming disadvantage to the model since most companies pay dividends to their share holders. The second assumption is that European terms are used ,European exercise terms demands that that option be exercised only at an expiry date, while the American exercise term demands that the term option be exercised at any time during the life of the option, this difference makes the American options more valuable due to their flexibility. The third assumption is that markets are efficient, it shows that people can not easily predict the direction of the market, that the market operates continuously with the share prices following continuous process. The fourth assumption is that no commissions are charged, it is normal that the market participants to pay a commission to purchase or sell options. The charge that people pay as commissions can easily distort the output of the model thus it should not be ignored, it should be considered. The fifthy assumption is that interest rates remain constant and known, Black Scholes model uses this risk free interest rate to represent this constant and known rate, but in reality there is no such a thing like risk free rate but discount rate in US government hence violating one of the assumption of the model(/b-s_model.htm). Lastly, returns are normally distributed, the assumption suggest that returns on the underlying stock are normally distributed something which is advantageous to most assets that offer options. Black Scholes model is widely applied in stock and currency market since it helps the options traders to do some comparisons between the current option prices, in exchange with the theoretical value to analyze weather a particular option contract is over or under valued, hence helping them to make wise decisions concerning various options. Before the invention of this model, no standard measure of option pricing method was there to calculate the options value hence no one was able to calculate and predict accurately prices for options (/call-option-price-formula-). But after the invention of Black Scholes model, it was easy for any one to calculate the option value into a reality, hence turning the market into a profitable business opportunity for investors. The application of the Black Scholes model in stock and currency market is done as explained in the following paragraphs. There are two methods of pricing option and put up option on the stock and currency market; option pricing and warrant pricing. In warrant pricing, the call options are issued by the corporation, although they tend to have longer duration than to exchange traded call option. Warrants can be valued by Black Scholes formula but it needs some modification in some of the parameters used (/finance/black-scholes/). When using Black Scholes model to value the warrants, it is advisable to use total amounts instead of per share amounts. The term warrant literally means to give or endow the right of owning something to some body else, warrants are always put to bonds or preferred stock as a bait to allow the issuer to pay lower interest rates or dividends, warrants can also be used in private equity deals, oftenly this warrant are not attached to bonds or preferred stock and can be sold independently from the stock or bonds. For warrants issued with preferred stocks, the holders detach them and sell before they can receive the payment of the dividends. We have various types of warrants; the traders have different reasons why they opt to invest in a particular type warrant than the other. We have equity warrants which are composed of callable and putable warrants. Call warrant give the right to buy the underlying securities while put up warrants gives someone a right to sell the underlying securities. Covered warrant is a warrant that has some underlying backing since the issuer will purchase the stock using other instruments to cover the option. Basket warrants are warrants that show the performance of the industry. Wedding warrants are warrants that are attached to the host debentures and are exercised only if the host debentures are returned back. Detachable warrants are warrants that are always detached from the debenture and traded separately. Lastly we have the naked warrants are warrants that are issued without accompanying bond and are traded on the stock exchange market. Warrants have certain characteristics which need to be considered by the traders when operating with them. Warrant premium shows how much extra one has to pay for his shares when purchasing them through the warrant as compared to purchasing them in a normal way. A warrant gearing is how to determine how much exposure one has to the underlying shares using the warrant as compared to the exposure he would have when if he purchases the share through the market. Expiration date is also one of the characteristics of warrants, this is the date the warrant expires, and one has to exercise the warrant before the expiration date. The last characteristic of warrant is the restriction on exercise, there are different exercises and the holder can only exercise on expiration date. Option pricing is one of the modern pricing techniques always considered among the mathematically complex of all applied areas of finance. It is used by the analysts where they are now able to calculate with high accuracy the value of the stock option. The purpose of pricing option is to ascertain the theoretical value for call or put option given certain known parame...